Sumsets via Spectral Properties of Dynamical Systems John Griesmer - - PowerPoint PPT Presentation

Sumsets via Spectral Properties of Dynamical Systems

John Griesmer

University of Denver

Ergodic Theory with Connections to Arithmetic University of Crete June 4, 2013

John Griesmer (U Denver) Sumsets via Spectral Properties June 4 1 / 19

These slides are mostly based on the article Sumsets of dense sets and sparse sets, Israel Journal of Mathematics, Volume 90, issue 1, pp 229-252, doi 10.1007/s11856-012-0008-1, preprint at arXiv:0911.2278

John Griesmer (U Denver) Sumsets via Spectral Properties June 4 2 / 19

Definitions and notation

The upper Banach density of E ⊂ Z is d∗(E) := lim

k→∞

sup

|I|=k I an interval

|E ∩ I| |I| If F ⊂ Z is finite, we say the finite configuration F appears in E if F + t ⊂ E for some t ∈ Z. Observation: the upper Banach density of E depends only on the finite configurations appearing in E.

John Griesmer (U Denver) Sumsets via Spectral Properties June 4 3 / 19

Szemer´ edi’s Theorem

Theorem (Szemer´ edi)

If E ⊂ Z has d∗(E) > 0, then E contains arithmetic progressions of every finite length. Equivalent to

Theorem (Furstenberg)

If (X, µ, T) is a probability measure preserving system and D ⊂ X has µ(D) > 0, then for all k, there exists n > 0 such that µ(D ∩ T −nD ∩ T −2nD ∩ · · · ∩ T −(k−1)nD) > 0 Many generalizations and refinements followed.

John Griesmer (U Denver) Sumsets via Spectral Properties June 4 4 / 19

Refinements of Szemer´ edi’s Theorem

A prototypical refinement:

Theorem (Bergelson and Leibman)

If d∗(E) > 0 and k ∈ N, then E contains a set of the form {a, a + n2, a + 2n2, · · · , a + (k − 1)n2}. Continuing problem: what kinds finite configurations necessarily appear in sets E having d∗(E) > 0? Many positive results, due to: Austin, Balister, Bergelson, Chu, Conze, Frantzikinakis, Furstenberg, Green, Host, Katznelson, Kra, Leibman, Lesigne, McCutcheon, Tao, Wierdl, Ziegler, and Zorin-Kranich, in various combinations.

John Griesmer (U Denver) Sumsets via Spectral Properties June 4 5 / 19

Sumsets

We now consider sets of the form A + B := {a + b : a ∈ A, b ∈ B}, where A, B ⊂ Z. (The sumset of A and B) What finite configurations must appear in A + B? When d∗(A) > 0 and d∗(B) > 0, there is a precise answer.

Definition

A set S ⊂ Z is a Bohr set if there is

1 A compact abelian group Z 2 An open set U ⊂ Z 3 An α in Z such that {nα}n∈Z is dense in Z

such that S ⊃ {n : nα ∈ U}. A set E is piecewise Bohr (PW-Bohr) if there is a Bohr set S such that E contains all the finite configurations contained in S; meaning for all finite F ⊂ S, F + t ⊂ E for some t.

John Griesmer (U Denver) Sumsets via Spectral Properties June 4 6 / 19

examples of Bohr and PW-Bohr sets

Definition

A set S ⊂ Z is a Bohr set if there is

1 A compact abelian group Z 2 An open set U ⊂ Z 3 An α in Z such that {nα}n∈Z is dense in Z

such that S ⊃ {n : nα ∈ U}. A set E is piecewise Bohr (PW-Bohr) if there is a Bohr set S such that E contains all the finite configurations contained in S; meaning for all finite F ⊂ S, F + t ⊂ E for some t. S:={n : n √ 2 mod 1 ∈ (0, 1/4)} is a Bohr set with Z = R/Z, U = (0, 1/4), and α = √ 2. E := S ∩ ∞

n=1[n2, n2 + n] is piecewise Bohr.

John Griesmer (U Denver) Sumsets via Spectral Properties June 4 7 / 19

Sumsets with dense summands are PW-Bohr

Theorem (Bergelson, Furstenberg, Weiss, 2006)

If d∗(A) > 0 and d∗(B) > 0 then A + B is PW-Bohr. Generalized to amenable groups by Bergelson, Beiglb¨

• ck, and Fish.

Strengthens Jin’s Theorem, which says that such A + B are piecewise syndetic. Jin’s Theorem was generalized to Zd by Jin and Keisler. Ultrafilter proof due to Beiglb¨

• ck.

Very elegant, very general approach forthcoming due to Bj¨

• rklund and

Fish.

Theorem ( Beiglb¨

• ck, Bergelson, Fish, 2009)

If S is PW-Bohr, then S ⊃ A + B where d∗(A) > 0 and d∗(B) > 0.

John Griesmer (U Denver) Sumsets via Spectral Properties June 4 8 / 19

When d∗(B) > 0 and d∗(A) = 0

Let S = {⌊n5/2⌋ : n ∈ Z}. Define the upper density relative to S: dS(A) := lim sup

N→∞

|A ∩ S ∩ [1, . . . N]| |S ∩ [1, . . . N]|

Main Theorem (G., 2009)

If dS(A) > 0 and d∗(B) > 0, then A + B is piecewise Bohr.

John Griesmer (U Denver) Sumsets via Spectral Properties June 4 9 / 19

Outline of the proof, I

• I. Translate to a statement about measure preserving systems: standard

Furstenberg correspondence principle argument. We omit the precise statement of the correspondence principle. Let (X, µ, T) be an ergodic probability measure preserving system. Fix A = {a1, a2, a3, . . . } ⊂ Z with dS(A) > 0 and D ⊂ X with µ(D) > 0. Study unions

• a∈A

T aD. (Suggested to me by M. Bj¨

• rklund and A. Fish.)

Since f := 1D is supported on D, the averages FN := 1 N

N

• n=1

f ◦ T −an are supported on

a∈A T aD.

John Griesmer (U Denver) Sumsets via Spectral Properties June 4 10 / 19

Outline of proof, II

• II. Reduce from general measure preserving systems to group rotations.

Key fact: for all θ ∈ (0, 1) lim

N→∞

1 N

N

• n=1

exp(2πiθ⌊n5/2⌋) = 0 (not just irrational θ) In the limit of FN := 1 N

N

• n=1

f ◦ T −an

• nly the eigenfunctions of T matter.

We need only consider ergodic group rotations.

John Griesmer (U Denver) Sumsets via Spectral Properties June 4 11 / 19

Outline of proof, III

• III. Identify lim

N→∞

1 N

N

• n=1

f ◦ T −an when (X, µ, T) = (Z, m, R), where

• Z is a compact abelian group with Haar measure m
• R : Z → Z is given by Rz = z + α.

Think of the average as a convolution of measures νN ∗ (f dm) where νN = 1 N

N

• n=1

δanα, Now ν := limN→∞ νN ≪ m, and ν(Z) = dS(A) (use equidistribution) So νN ∗ (f dm) → ν ∗ f dm, convolution of two functions in L2(m), continuous.

John Griesmer (U Denver) Sumsets via Spectral Properties June 4 12 / 19

Outline of proof, IV

• IV. So νN ∗ (f dm) → ν ∗ f dm, convolution of two functions in L2(m).

We have shown F := lim

N→∞

1 N

N

• n=1

f ◦ T −an is continuous (actually F = a continuous function m-a.e.) So

a∈A T aD contains an open set (up to m-measure 0)

The correspondence principle turns this into the desired conclusion.

• John Griesmer (U Denver)

Sumsets via Spectral Properties June 4 13 / 19

Perfect squares vs ⌊n5/2⌋.

You can replace ⌊n5/2⌋ with any increasing sequence cn having the equidistribution property and get the same conclusion. Equidistribution of cn means lim

n→∞

1 N

N

• n=1

exp(2πiθcn) = 0 for all θ ∈ (0, 1). What about Q = {n2 : n ∈ N}? Not equidistributed – take θ = 1

4.

Is Q + B PW-Bohr whenever d∗(B) > 0? No:

Example

There is a set B ⊂ Z having d∗(B) > 1 − ε such that Q + B is not PW-Bohr (not even piecewise syndetic).

John Griesmer (U Denver) Sumsets via Spectral Properties June 4 14 / 19

Example

There is a set B ⊂ Z having d∗(B) > 1 − ε such that Q + B is not PW-Bohr (not even piecewise syndetic). Construction: Let Z = Z/2Z × Z/3Z × Z/5Z × · · · , the direct product of the cyclic groups of prime order. Let α = (1, 1, 1, 1, . . . ), so that nα is dense in Z. Now Q := {n2α : n ∈ Z} has Haar measure 0. There is a compact K ⊂ Z such that m(K) = 1 − ε and Q + K has empty interior. Use the ergodic theorem to copy this behavior to Z.

• John Griesmer (U Denver)

Sumsets via Spectral Properties June 4 15 / 19

Possible generalizations

The proof of the main theorem used equidistribution twice: (i) first to reduce from general measure preserving systems to Kronecker systems, (ii) then to estimate the measure of A := {nα : n ∈ A}. The perfect squares example shows that we need m( A) > 0 to conclude that A + B is PW-Bohr whenever d∗(B) > 0. Is this all we need?

Question

Suppose A ⊂ Z satisfies: Aα is dense in Z whenever Zα is dense in Z, meaning A is dense in the Bohr topology of Z. Does this imply d∗(A + B) = 1 whenever d∗(B) > 0?

John Griesmer (U Denver) Sumsets via Spectral Properties June 4 16 / 19

Possible generalizations

Question

Suppose A ⊂ Z satisfies: Aα is dense in Z whenever Zα is dense in Z, meaning A is dense in the Bohr topology of Z. Does this imply d∗(A + B) = 1 whenever d∗(B) > 0? Why this is hard: the hypothesis on A does not imply equidistribution.

Example (Katznelson ’73)

There is an atomless Borel probability measure σ on [0, 1] such that A :=

• n :
• exp(2πinθ) dσ(θ)
• > 1

2

• is dense in the Bohr topology on Z.

John Griesmer (U Denver) Sumsets via Spectral Properties June 4 17 / 19

Sets of measurable recurrence and sumsets

We say that S ⊂ Z is a set of measurable recurrence if for every probability M.P.S. (X, µ, T) and every D ⊂ X with µ(D) > 0, there exists n ∈ S such that µ(D ∩ T −nD) > 0. We say that A ⊂ Z is measurably generating if for every ergodic probability M.P.S. (X, µ, T) and every D ⊂ X, µ

a∈A

T aD

• = 1.

Fact: A is measurably generating if and only if A + t is a set of measurable recurrence for every t.

Question

If A is dense in the Bohr topology of Z, is A a set of measurable recurrence? (Asked by Bergelson and Ruzsa, 2009)

John Griesmer (U Denver) Sumsets via Spectral Properties June 4 18 / 19

Acknowledgements

Thanks to Michael Bj¨

• rklund and Alexander Fish.

John Griesmer (U Denver) Sumsets via Spectral Properties June 4 19 / 19